Boundary CM points and class groups of small exponent

David Aiken; Dmitriy Shvydkoy; Erick Ross; Hui Xue

arxiv: 2605.26020 · v1 · pith:VMWBC67Onew · submitted 2026-05-25 · 🧮 math.NT

Boundary CM points and class groups of small exponent

David Aiken , Erick Ross , Dmitriy Shvydkoy , Hui Xue This is my paper

Pith reviewed 2026-06-29 20:12 UTC · model grok-4.3

classification 🧮 math.NT

keywords CM pointsequidistributionfundamental domainclass groupsnegative discriminantsboundaryquadratic ordersideal class group exponent

0 comments

The pith

CM points of negative discriminants equidistribute on the boundary of the fundamental domain.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Duke showed that CM points of negative discriminants equidistribute throughout the fundamental domain for SL_2(Z) as the discriminant tends to negative infinity. This paper establishes the same equidistribution when the points are restricted to the boundary of that domain. It supplies a complete characterization of the discriminants for which every CM point lies on the boundary. It also supplies a conditional classification of all negative discriminants whose class groups have small exponent.

Core claim

The sets CM_D become equidistributed on the boundary of the fundamental domain as D tends to negative infinity. The discriminants D for which every CM_D point lies on the boundary are completely characterized. Conditionally, all negative discriminants with class group of small exponent are classified.

What carries the argument

The sets of CM_D points lying on the boundary arcs of the fundamental domain, whose distribution is governed by the class group of the corresponding quadratic order.

If this is right

The proportion of CM points lying on the boundary admits an explicit asymptotic density.
Only discriminants belonging to a specific arithmetic progression or list have the property that every CM point lies on the boundary.
The class groups of small exponent occur for only finitely many or explicitly listed negative discriminants under the assumption.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Boundary equidistribution may be studied with methods that do not require control over interior points.
The same circle of ideas could apply to equidistribution questions for other Fuchsian groups or for points in higher-dimensional symmetric spaces.
Progress on the unspecified assumption would turn the classification into an unconditional theorem.

Load-bearing premise

The classification of negative discriminants with small-exponent class groups holds only under an unspecified assumption.

What would settle it

A sequence of negative discriminants for which the corresponding CM points on the boundary fail to equidistribute, or a negative discriminant with small class-group exponent lying outside the claimed list.

read the original abstract

Let $\mathcal F$ denote the fundamental domain for $\text{SL}_2(\mathbb{Z})$ on the upper half plane $\mathcal H$. William Duke showed that as fundamental discriminants $D \to -\infty$, the sets $\mathrm{CM}_{D}$ (CM points of discriminant $D$) are equidistributed in $\mathcal F$. In this paper, we investigate the behavior of CM points on the boundary of $\mathcal F$. We prove that such CM points are equidistributed on the boundary, and also give a complete characterization of when every $\mathrm{CM}_D$ point lies on the boundary. Along the way, we also (conditionally) give a complete classification of negative discriminants with class group of small exponent.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The boundary equidistribution and characterization of boundary CM points are the real new pieces here, while the conditional classification of small-exponent class groups is harder to judge without the assumption stated.

read the letter

The paper extends Duke's equidistribution of CM points to the boundary of the fundamental domain and gives a characterization of when every CM_D point lies on that boundary. It also supplies a conditional list of negative discriminants whose class groups have small exponent.

The boundary equidistribution and the characterization look like genuine additions. They are not direct corollaries of the cited Duke result and address a natural question about the behavior on the edge of the domain. The work builds on Duke without obvious circularity.

The conditional classification is the weaker element. The abstract marks it as conditional but gives no statement of the assumption. Without that, it is impossible to tell whether the condition is routine (such as GRH) or something narrower, and therefore how much weight the resulting list carries. If the paper spells out the assumption and the reduction in the body, this concern shrinks.

The paper targets people working on CM points, equidistribution on hyperbolic surfaces, or computational class group questions. A reader already following Duke-type results or needing explicit lists of discriminants with restricted class groups would find the boundary parts and the characterization directly useful.

I would send this to peer review. The core boundary claims are grounded in prior work and stated as new, so the manuscript deserves referee attention even if the conditional section needs more detail.

Referee Report

1 major / 1 minor

Summary. The paper extends Duke's theorem on equidistribution of CM points of negative fundamental discriminants D in the fundamental domain F for SL_2(Z). It proves equidistribution of these points on the boundary of F as D → -∞, gives a complete (unconditional) characterization of discriminants D for which every CM_D point lies on the boundary, and conditionally classifies all negative discriminants whose class groups have small exponent.

Significance. The boundary equidistribution result is a natural extension of Duke's theorem and would strengthen understanding of CM point distributions if the proof holds. The unconditional characterization of boundary-lying CM_D points is a precise and potentially useful contribution. The conditional classification of small-exponent class groups is of interest in class field theory but its significance depends on the (unstated in the abstract) assumption; if that assumption is standard and the reduction direct, the result would be a solid addition.

major comments (1)

[Abstract] Abstract: the conditional classification of negative discriminants with small-exponent class groups is presented without stating the precise assumption. Since this claim is part of the paper's results, the assumption must be explicitly formulated (e.g., in §1 or the relevant theorem statement) so that its strength and the validity of the implication can be checked.

minor comments (1)

[§1] The introduction should include a brief statement of the assumption used for the classification result even if the full details appear later.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation of minor revision. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the conditional classification of negative discriminants with small-exponent class groups is presented without stating the precise assumption. Since this claim is part of the paper's results, the assumption must be explicitly formulated (e.g., in §1 or the relevant theorem statement) so that its strength and the validity of the implication can be checked.

Authors: We agree that the assumption should be stated explicitly so that readers can assess its strength. The assumption is formulated in the body of the paper (in the statement of the relevant theorem), but we acknowledge that the abstract only indicates the result is conditional without naming the assumption. We will revise the abstract (and, if needed, the introduction) to include a precise formulation of the assumption. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results extend Duke's theorem independently.

full rationale

The paper's core claims rest on Duke's equidistribution theorem (external, different author) for the main equidistribution result and boundary characterization. The conditional classification of discriminants with small-exponent class groups is flagged as depending on an external assumption whose statement is absent, but no derivation step reduces by construction to the paper's own inputs, fitted parameters, or self-citation chains. No self-definitional loops, renamed known results, or load-bearing self-citations appear in the provided abstract or described structure. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; the paper relies on standard facts from the theory of quadratic fields, SL_2(Z) fundamental domains, and equidistribution techniques, but specific axioms and parameters cannot be extracted without the full text.

axioms (1)

standard math Standard properties of the fundamental domain for SL_2(Z) on the upper half-plane
Used to define the boundary and CM points

pith-pipeline@v0.9.1-grok · 5656 in / 1180 out tokens · 29272 ms · 2026-06-29T20:12:37.803731+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 1 canonical work pages

[1]

Transcendence of zeros of modular forms.https: //erick-ross.github.io/research/Preprint, 2025

David Aiken, Erick Ross, Dmitriy Shvydkoy, and Hui Xue. Transcendence of zeros of modular forms.https: //erick-ross.github.io/research/Preprint, 2025

2025
[2]

Cox.Primes of the Formx 2 +ny 2: Fermat, Class Field Theory, and Complex Multiplication

David A. Cox.Primes of the Formx 2 +ny 2: Fermat, Class Field Theory, and Complex Multiplication. Wiley- Interscience, New York, 1989

1989
[3]

W. Duke. Hyperbolic distribution problems and half-integral weight Maass forms.Invent. Math., 92(1):73–90, 1988

1988
[4]

Imaginary quadratic number fields with class groups of small exponent.Acta Arith., 193(3):217–233, 2020

Andreas-Stephan Elsenhans, J¨ urgen Kl¨ uners, and Florin Nicolae. Imaginary quadratic number fields with class groups of small exponent.Acta Arith., 193(3):217–233, 2020

2020
[5]

Extremal elasticity of quadratic orders, 2025.https://arxiv.org/abs/2503.07801

Steve Fan and Paul Pollack. Extremal elasticity of quadratic orders, 2025.https://arxiv.org/abs/2503.07801

work page arXiv 2025
[6]

Zuckerman, and Hugh L

Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery.An introduction to the theory of numbers. John Wiley & Sons, Inc., New York, Fifth edition, 1991

1991
[7]

F. K. C. Rankin and H. P. F. Swinnerton-Dyer. On the zeros of Eisenstein series.Bull. London Math. Soc., 2:169–170, 1970

1970
[8]

Sarah Reitzes, Polina Vulakh, and Matthew P. Young. Zeros of certain combinations of Eisenstein series.Math- ematika, 63(2):666–695, 2017

2017
[9]

Class groups of small exponent.https://github.com/erick-ross/research-data/tree/master/ class-groups-small-exp, 2025

Erick Ross. Class groups of small exponent.https://github.com/erick-ross/research-data/tree/master/ class-groups-small-exp, 2025

2025
[10]

Zeros of cuspidal projections of products of Eisenstein series.Ramanujan J., 68(4):Pa- per No

Hui Xue and Daozhou Zhu. Zeros of cuspidal projections of products of Eisenstein series.Ramanujan J., 68(4):Pa- per No. 107, 49, 2025. 12 (D. Aiken)Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana (E. Ross)School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC Email address:erickjohnross@gmail.com (D....

2025

[1] [1]

Transcendence of zeros of modular forms.https: //erick-ross.github.io/research/Preprint, 2025

David Aiken, Erick Ross, Dmitriy Shvydkoy, and Hui Xue. Transcendence of zeros of modular forms.https: //erick-ross.github.io/research/Preprint, 2025

2025

[2] [2]

Cox.Primes of the Formx 2 +ny 2: Fermat, Class Field Theory, and Complex Multiplication

David A. Cox.Primes of the Formx 2 +ny 2: Fermat, Class Field Theory, and Complex Multiplication. Wiley- Interscience, New York, 1989

1989

[3] [3]

W. Duke. Hyperbolic distribution problems and half-integral weight Maass forms.Invent. Math., 92(1):73–90, 1988

1988

[4] [4]

Imaginary quadratic number fields with class groups of small exponent.Acta Arith., 193(3):217–233, 2020

Andreas-Stephan Elsenhans, J¨ urgen Kl¨ uners, and Florin Nicolae. Imaginary quadratic number fields with class groups of small exponent.Acta Arith., 193(3):217–233, 2020

2020

[5] [5]

Extremal elasticity of quadratic orders, 2025.https://arxiv.org/abs/2503.07801

Steve Fan and Paul Pollack. Extremal elasticity of quadratic orders, 2025.https://arxiv.org/abs/2503.07801

work page arXiv 2025

[6] [6]

Zuckerman, and Hugh L

Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery.An introduction to the theory of numbers. John Wiley & Sons, Inc., New York, Fifth edition, 1991

1991

[7] [7]

F. K. C. Rankin and H. P. F. Swinnerton-Dyer. On the zeros of Eisenstein series.Bull. London Math. Soc., 2:169–170, 1970

1970

[8] [8]

Sarah Reitzes, Polina Vulakh, and Matthew P. Young. Zeros of certain combinations of Eisenstein series.Math- ematika, 63(2):666–695, 2017

2017

[9] [9]

Class groups of small exponent.https://github.com/erick-ross/research-data/tree/master/ class-groups-small-exp, 2025

Erick Ross. Class groups of small exponent.https://github.com/erick-ross/research-data/tree/master/ class-groups-small-exp, 2025

2025

[10] [10]

Zeros of cuspidal projections of products of Eisenstein series.Ramanujan J., 68(4):Pa- per No

Hui Xue and Daozhou Zhu. Zeros of cuspidal projections of products of Eisenstein series.Ramanujan J., 68(4):Pa- per No. 107, 49, 2025. 12 (D. Aiken)Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana (E. Ross)School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC Email address:erickjohnross@gmail.com (D....

2025